Asian Journal of Mathematics and Computer Research

This paper establishes a general theorem on absolute matrix summability for infinite series by employing the class of quasi-f-power increasing sequences in place of almost increasing sequences. The study is framed within the φ − |B, p_n|k summability method associated with a positive normal matrix B = (bnv) and a sequence of positive coefficients (pn). After recalling the relevant Riesz mean, normal matrix transformations, and bounded variation assumptions, the paper states a theorem that extends Bor’s result on absolute Riesz summability factors. The theorem assumes standard monotonicity and boundedness conditions on the matrix entries, the sequence (λ_n), the auxiliary sequence (β_n), and the n-th Ces`aro mean (t_n) of (na_n). Under these conditions, it is shown that the transformed series  a_nλ_n is summable by the φ−|B, p_n|k method for k ≥ 1. The proof proceeds through Abel’s transformation, decomposes the matrix transform into four terms, and establishes convergence estimates for each term by using H¨older’s inequality, the properties of the matrix, and the consequences of quasi-f-power increasing sequences. The result confirms that replacing almost increasing sequences with the broader quasi-f-power increasing framework preserves the required summability conclusion. Several special cases follow from the theorem, including results for quasi-δ-power increasing sequences, | ¯N , p_n|k summability, and |B, p_n|k summability under appropriate choices of parameters. Thus, the theorem provides a unified formulation for related absolute summability factor results within the stated hypotheses. No additional assumptions are introduced beyond those specified in the theorem, and the derived consequences are presented only as formal reductions of the main result. This maintains a close connection between the generalised theorem and the earlier summability factor results considered in the manuscript.